International Journal of Wavelets, Multiresolution and Information Processing

This paper considers the recovery condition of signals from undersampled data corrupted with additive noise in the framework of cumulative coherence. We establish some sufficient conditions which can guarantee the stable recovery from quadratically constrained basis pursuit (QCBP), Dantzig selector (DS) and Lasso estimator.

The offset linear canonical transform (OLCT) has proven to be a novel and effective method in signal processing and optics. Many important properties and results of the OLCT have been well studied and published. In this work, the sampling theorem of the OLCT bandlimited signals based on reproducing kernel Hilbert space has been proposed. First, we show that the bandlimited signals in the OLCT domain form a reproducing kernel Hilbert space. Then, an orthogonal basis for the OLCT bandlimited signals has been obtained based on the reproducing kernel Hilbert space. By using the orthogonal basis, the uniform sampling theory for bandlimited signals associated with the OLCT has been obtained. Furthermore, the nonuniform sampling of the OLCT bandlimited signals also has been attained. Finally, the simulations are provided to prove the usefulness and correctness of the derived results.

The main purpose of this paper is to provide a characterization of scaling functions for non-uniform multiresolution analysis (NUMRA, in short). Some necessary and sufficient conditions for scaling functions of wavelet NUMRA in the frequency domain are also obtained.

This paper considers recovery of matrices that are low rank or approximately low rank from linear measurements corrupted with additive noise. We study minimization of the difference of Nuclear and Frobenius norms (abbreviated as $S_{1,2}$ norm) as a nonconvex and Lipschitz continuous metric for solving this noisy low rank matrix recovery problem. We mainly study two types of bounded observation noisy low rank matrix recovery problems, including the $l_2$-norm bounded noise and the Dantizg Selector noise. Based on the matrix restricted isometry property (abbreviated as M-RIP), we prove that this $S_{1,2}$ norm-based minimization method can stably recover a (approximately) low rank matrix in the two types bounded noisy low rank matrix recovery problems. In addition, we use the truncated difference of Nuclear and Frobenius norms (denoted as the truncated $S_{1,2}$ norm) to recover a low rank matrix when the observation noise is the Dantizg Selector noise. We give the stable recovery result for this truncated $S_{1,2}$ norm minimization in Dantizg Selector noise case when the linear measurement map satisfies the M-RIP condition.

Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed.

Multiple kernel learning (MKL) aims at learning an optimal combination of base kernels with which an appropriate hypothesis is determined on the training data. MKL has its flexibility featured by automated kernel learning, and also reflects the fact that typical learning problems often involve multiple and heterogeneous data sources. Target kernel is one of the most important parts of many MKL methods. These methods find the kernel weights by maximizing the similarity or alignment between weighted kernel and target kernel. The existing target kernels implement a global manner, which (1) defines the same target value for closer and farther sample pairs, and inappropriately neglects the variation of samples; (2) is independent of training data, and is hardly approximated by base kernels. As a result, maximizing the similarity to the global target kernel could make these pre-specified kernels less effectively utilized, further reducing the classification performance. In this paper, instead of defining a global target kernel, a localized target kernel is calculated for each sample pair from the training data, which is flexible and able to well handle the sample variations. A new target kernel named empirical target kernel is proposed in this research to implement this idea, and three corresponding algorithms are designed to efficiently utilize the proposed empirical target kernel. Experiments are conducted on four challenging MKL problems. The results show that our algorithms outperform other methods, verifying the effectiveness and superiority of the proposed methods.

Savitzky–Golay (S-G) filter is a method of local polynomial regression, and iterative filtering with S-G filter can be used to smooth out random noise and outliers of cloud noise in NDVI time series. It involves a continuous approximation to the upper envelope of NDVI time series. In this paper, the optimum-length of S-G filter was estimated based on Steinc’s unbiased risk estimator theory when S-G filtering was conducted iteratively, and the reconstruction result was presented. Reconstruction experiments on the simulated data and MODIS NDVI time series of the year 2010–2014 showed that the optimum-length S-G filter can outperform the fixed bandwidth S-G filter.

Let $L^2(ℝ,ℍ)$ be a space of square integrable quaternionic-valued functions defined on real line. In this paper, for $\mathrm{\Phi }\in L^2(ℝ,ℍ),$ $a>1$ and $q>0$, if the sequence of functions $\{a^{m/2}\mathrm{\Phi }(a^{m}x-nq),m,n\in \mathbb{Z}\}$ is a wavelet frame of $L^2(ℝ,ℍ)$, we study the stability of the wavelet frame when the sampling $\{n\}$ or the mother wavelet $\mathrm{\Phi }$ has perturbation by using the theory of wavelet analysis.

Facial expression recognition is one of the hotspots in the fields of computer vision and deep learning. It has very important applications in the domains of learning service recommendation, human–computer interaction and medical industry. Aiming at the problem that the traditional expression recognition method is not accurate, this paper proposes a method combining Gabor wavelet transform and convolutional neural network. Firstly, face positioning, cropping, histogram equalization and other preprocessing are performed on the expression image. Then we extract key frames of expression sequences. After that the Gabor wavelet transform is performed on the expression image to obtain magnitude and phase characteristics. Finally, we design a 2-channel CNN for training and classification. The experiment achieves an accuracy of 96.81% on the CK+ database and it has a certain improvement compared with the Gabor wavelet transform and the traditional CNN alone.

In this paper, it is proved that under some conditions, wavelet packet basis of $L^2(ℝ)$ can be used as a tool for the uniform approximation of an $M$-times ($M>0$) continuously differentiable and square integrable function $f$. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function $f$ at lower levels are uniformly reliable and they uniformly approach zero as $j\to \infty$ are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in $L^2(ℝ)$ is given and its error analysis has been discussed.

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

Medical data classification is the process of transforming descriptions of medical diagnoses and procedures into universal medical code numbers. The diagnoses and procedures are usually taken from a variety of sources within the healthcare record, such as the transcription of the physician’s notes, laboratory results, radiologic results and other sources. However, there exist many frequency distribution problems in these domains. Hence, this paper intends to develop an advanced and precise medical data classification approach for diabetes and breast cancer dataset. With the knowledge of the features and challenges persisting with the state-of-the-art classification methods, deep learning-based medical data classification methodology is proposed here. It is well known that deep learning networks learn directly from the data. In this paper, the medical data is dimensionally reduced using Principle Component Analysis (PCA). The dimensionally reduced data are transformed by multiplying by a weighting factor, which is optimized using Whale Optimization Algorithm (WOA), to obtain the maximum distance between the features. As a result, the data are transformed into a label-distinguishable plane under which the Deep Belief Network (DBN) is adopted to perform the deep learning process, and the data classification is performed. Further, the proposed WOA-based DBN (WOADBN) method is compared with the Neural Network (NN), DBN, Generic Algorithm-based NN (GANN), GADBN, Particle Swarm Optimization (PSONN), PSO-based DBN (PSODBN), WOA-based NN (WOANN) techniques and the results are obtained, which shows the superiority of proposed algorithm over conventional methods.